WASHINGTON — Senate Republicans on Tuesday officially abandoned the latest plan to repeal the Affordable Care Act, shelving a showdown vote on the measure and effectively admitting defeat in their last-gasp drive to fulfill a core promise of President Trump and Republican lawmakers.

“We haven’t given up on changing the American health care system,” Senator Mitch McConnell of Kentucky, the majority leader, said after a lunchtime meeting of Republican senators. “We are not going to be able to do that this week, but it still lies ahead of us, and we haven’t given up on that.”

Mr. McConnell said Republicans would move on to their next big legislative goal: overhauling the tax code, a feat that has not been accomplished since 1986.

Democrats, who have spent all year fighting to protect the Affordable Care Act, a law that is a pillar of President Barack Obama’s legacy, responded by calling for the resumption of bipartisan negotiations to stabilize health insurance markets. Republican leaders had squelched those talks as the latest repeal plan, written by Senators Lindsey Graham of South Carolina and Bill Cassidy of Louisiana, gained steam.

“We hope we can move forward and improve health care, not engage in another battle to take it away from people, because they will fail once again if they try,” said Senator Chuck Schumer of New York, the Democratic leader.

The decision by Senate Republican leaders may prove to be a milestone in the decades-long fight over health insurance in the United States, suggesting that the Affordable Care Act had gained at least a reprieve and perhaps a measure of political acceptance.

Senator Lamar Alexander, Republican of Tennessee and the chairman of the Senate health committee, and Senator Patty Murray of Washington, the senior Democrat on the panel, have been working on legislation to stabilize insurance markets and hold down premiums in the next couple of years. Both said on Tuesday that they hoped to resume those efforts.

Millions of people who buy insurance on their own face sharp increases in premiums next year, and Trump administration officials have taken a number of steps that have already undermined the operations of the health law.

And health care is sure to be an issue in next year’s midterm elections.

Senator John Thune of South Dakota, a member of the Senate Republican leadership, said the concept behind the Graham-Cassidy bill would help Republicans define their differences with Democrats in the campaign season.

“Single payer, socialism — or federalism, returning power to the states to make decisions,” Mr. Thune said. “I think that’s a great contrast for us, and I think that’s an argument eventually we can win.”

For their part, Democrats have tried to use health care as a bludgeon against the few Senate Republican targets they have next year, mainly Senators Dean Heller of Nevada and Jeff Flake of Arizona.

“We know Republicans like Dean Heller and Jeff Flake won’t stop until they force Americans to pay more for less, and we will make sure voters hold them accountable for it,” the Democratic Senatorial Campaign Committee said in a statement this week.

The Graham-Cassidy bill would have taken money provided under the Affordable Care Act for insurance subsidies and the expansion of Medicaid and sent it to states in the form of block grants.

The bill’s demise was welcomed by consumer groups, doctors, hospitals and insurance executives who mobilized opposition to the proposal. The Congressional Budget Office said Monday that the bill would have reduced projected federal Medicaid spending by $1 trillion over a decade, added millions of Americans to the ranks of the uninsured and eliminated consumer protections for some people with pre-existing conditions.

This time, Senate Republicans were trying to pass a different proposal, and a deadline was fast approaching: They had only until the end of this week to pass a repeal bill using special budget rules that shield it from a Democratic filibuster.

Mr. McConnell had planned for a vote before that deadline, but he could afford to lose only two Republicans. By Tuesday, three members of his party had already gone public with their firm opposition: Ms. Collins, Mr. McCain and Senator Rand Paul of Kentucky.

None of the three senators seemed likely to change their minds. Mr. McCain detested the hasty, partisan process used to push the bill; Ms. Collins had broad concerns about the legislation’s effects on health care; and Mr. Paul objected to the fundamental architecture of the bill, which in his view constituted “fake repeal.”

Senator Orrin G. Hatch, Republican of Utah and the chairman of the Senate Finance Committee, said he saw no reason for the Senate to vote.

“If you don’t have the votes,” he said, “I don’t see why we would waste the time of the Senate.”

And other Republican senators might have opposed it had party leaders moved forward. In a statement released after Senate Republicans decided not to do so, Senator Lisa Murkowski of Alaska, who voted against the last repeal attempt in July, denounced what she described as “a lousy process.”

“The U.S. Senate cannot get the text of a bill on a Sunday night, then proceed to a vote just days later, with only one hearing — and especially not on an issue that is intensely personal to all of us,” Ms. Murkowski said, without saying which way she would have voted.

Mr. Cassidy was blunt: “We don’t have the votes,” he said, adding, “Am I disappointed? Absolutely.”

And Senator David Perdue, Republican of Georgia, was left fuming.

“I am kind of disgusted that after nine months, the self-interest is still outweighing the national interest in our caucus in some ways,” he said.

But Mr. Graham still held out the hope that the repeal proposal would pass — just later, after Republicans tackle taxes, and when Republicans can consider the repeal plan in a more deliberative fashion.

“There are 50 votes for the substance,” Mr. Graham said. “There are not 50 votes for the process.”

The tax effort will probably occupy Congress through the remainder of this year, and into next year.

After tackling the tax overhaul, Republicans could make another attempt at passing a health bill without needing any Democratic votes. But such an undertaking would require passing yet another budget blueprint, in order to protect the bill from a Democratic filibuster, and it would put health care front and center as lawmakers head into the midterm elections.

Even as Republicans shift their focus to taxes, pressure is building for a bipartisan response to the problems of the health care system. The Alexander-Murray stabilization package has two main elements: money for subsidies paid to insurance companies, to reimburse them for reducing out-of-pocket costs for low-income people, and more freedom for states to revamp their insurance markets.

Democrats see the cost-sharing subsidy payments as essential. But many Republicans, including Mr. Trump, have described them as a bailout for insurance companies that would prop up the Affordable Care Act.

“I think most of the members would suggest they really don’t know that they can stabilize the product at this stage of the game,” said Senator Mike Rounds, Republican of South Dakota. “I think there’s a number of them that are going to be very hesitant to try to put more money into a system that they are convinced is going south.”

Earlier Tuesday, with little hope of success in the Senate, Mr. Trump expressed his displeasure.

“At some point there will be a repeal and replace, but we’ll see whether or not that point is now or will it be shortly thereafter,” he said at the White House. “But we are disappointed in certain so-called Republicans.”